Friedrich Schneider: Equivariant dissipation in permutation groups

Data: segunda-feira, 25 de junho de 2018, às 10h00.

Sala: Auditório Jacy Monteiro

Friedrich Martin Schneider (UFSC)

Título: Equivariant dissipation in permutation groups.

Resumo: In his seminal work on metric measure geometry, Gromov introduced the observable distance, a metric on the set of isomorphism classes of metric measure spaces. This metric generalizes the well-known measure concentration phenomenon in a very natural way: a sequence of spaces has the Levy concentration property if and only if it converges to a singleton space with respect to the observable distance. Motivated by striking applications of measure concentration in topological dynamics, Vladimir Pestov proposed to study the phenomenon of concentration to non-trivial spaces in the context of topological groups. Prompted by work of Glasner and Weiss, in 2006 he posed the following problem: given a left-invariant metric d on the full symmetric group of the natural numbers, compatible with the topology of pointwise convergence, does the sequence of finite symmetric groups, equipped with their normalized counting measures and the restrictions of d, concentrate to the compact space of linear order on the natural numbers, endowed with its unique invariant Borel probability measure and a suitable compatible metric?

In the talk, I will answer Vladimir Pestov's question in the negative. As we will see, the above sequence of finite symmetric groups indeed dissipates (in the sense of Gromov), thus does not even admit a subsequence being Cauchy with respect to the observable distance.